to me, the more important factor is starting early enough to have cool temps for all that climbing from 8400ft warmth to 12,000 ft.
Then again, that might get you there at noon, and you have the rest of the day to sit around in the sun or ugh, hot tent at shadeless Trail Camp. Your choice.
I have never had a problem finding a campsite at TrailCamp. yes, lots of people there, but plenty of creative sites are located around the lake, off the trail, or at nearby Consultation Lake.